Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

“分段法”在极限和积分中的应用

举例说明“分段方法”在极限和积分中的应用．     

Classical fluids of negative heat capacity

It is shown that new parametersX can be defined such that the heat capacity C_xT(S/T)_x is negative, even when the canonical ensemble [i.e., at fixed T=(U/S)_Y and YX] is stable. This implies an extension of the classical theory of polytropes from ideal gases to general fluids. As examples of negative heat capacity systems we treat blackbody radiation and general gas systems with nonsingular _T. For the case of a simple ideal gas we even exhibit an apparatus which enforces a constraint X(p, V)=const that makes C_x<0. We then show that it is possible to infer the statistical mechanics of canonicallyunstable systems-for which even the traditional heat capacities are negative-by imposing constraints that stabilize the associated noncanonical ensembles. Two explicit models are discussed.     

Spherical models with a Gates-Penrose-type phase transition

Gates and Penrose have given criteria under which classical gases with weak long-range interactions fail to be described by the van der Waals equation with Maxwell's rule. Unfortunately, examples of equations of state for such systems have not yet been produced. This paper examines the Gates-Penrose class of interactions-i.e.,U (r)=q(r)+(r), in the limit0, where the Fourier transform (p) has a minimum at a nonzero value ofp-for the spherical model on a one-dimensional lattice. Free energy and magnetization isotherms are computed; it is seen that there is a phase transition, but that the zero-field spontaneous magnetization is always zero (a parahelicoidal phase). However, the pair-correlation function may exhibit either long-range order or the appearance of oscillation.     

Peculiarities of changes of some physicomechanical characteristics of monoisotopes B, B and natural β-boron

Real structure and some physicomechanical characteristics of the samples of natural beta-rhombohedral boron B as well as of its ¹⁰B and ¹¹B monoisotopes have been studied. It was shown that the influence of ¹⁰B and ¹¹B isotopes on physicomechanical properties of boron had a different character. In particular, the samples enriched with ¹¹B had high values of microhardness, shear modulus (SM) and elastic limit if compared to those of boron, while the samples enriched with ¹⁰B monoisotopes were characterized with high values of thermal expansion coefficient (TEC) and thermal conductivity; lattice parameters a and c increased by the sequence: ¹¹B, B, ¹⁰B. It was established that TEC, thermal conductivity, microhardness, SM and shear elastic limit increased in all samples at annealing for 5 h at 1500 °C regardless of isotope content.     

Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Summary The members of the power divergence family of statistics all have an asymptotically equivalent χ² distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX ² statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.     

Spectral density distribution moments of a Hamiltonian of N coupled Morse oscillators: The large‐N limit

The expressions for moments of spectral density distribution of a Hamiltonian representing a system of N coupled Morse oscillators are reduced to the large‐N limit. The effect of the values of coupling constants on the spectral density distribution is analyzed. For the values of coupling constants within the pertinent range for real molecules, the relations between moments appear to be close to those for a Gaussian distribution. Outside this range, the deviations from the normal distribution are substantial. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

Improving the lower detection limit of potentiometric sensors by covalently binding the ionophore to a polymer backbone

The lower detection limit of polymeric membrane ion-selective electrodes (ISEs) is impaired by zero-current ion fluxes through the organic phase. This adverse effect is largely eliminated by covalently attaching the ionophore to a polymer backbone. To this purpose, the Pb²⁺-selective ligand, 4-tert-butylcalix[4] arene-tetrakis(N,N′-dimethylthioacetamide) is substituted on its upper rim by a diol derivative which is subsequently copolymerized with poly(tetrahydrofuran)diol and 2,2,4-trimethylhexamethylene diisocyanate to the corresponding polyurethane. By measurements on sandwich membranes, it is shown that through binding the ionophore to the polymer, the mobility of Pb²⁺ in the ISE membrane is strongly reduced. As a consequence, the response range of such an ISE is extended by several orders of magnitude. This is the case even when using an internal electrolyte that with an ISE based on a mobile ionophore leads to strong deviations from the linear response because of ion uptake from the sample into the membrane or ion release from the membrane into the sample. With a conventional inner filling solution of 10⁻¹ M Pb(NO₃)₂, a lower detection limit of 1.7×10⁻⁹ M Pb²⁺ has been achieved in the presence of 10⁻⁴ M Na⁺.     

Derivation of the transport equation for electrons moving through random impurities

A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength. The random strength is given by a field of Gaussian random variables. We show the existence of the averaged dynamics and prove that in the weak coupling limit, 0, ₂ t= fixed, one obtains the usual transport equation for the velocity distribution.Work supported by a Max Kade Foundation fellowship.On leave of absence of the Fachbereich Physik der Universität München.